Constant Problem

Given an expression involving known constants, integration in finite terms, computation of limits, etc., determine if the expression is equal to Zero. The constant problem is a very difficult unsolved problem in Transcendental Number theory. However, it is known that the problem is Undecidable if the expression involves oscillatory functions such as Sine. However, the Ferguson-Forcade Algorithm is a practical algorithm for determining if there exist integers $a_i$ for given real numbers $x_i$ such that

$$a_1x_1+a_2x_2+\ldots+a_nx_n=0,$$

or else establish bounds within which no relation can exist (Bailey 1988).

See also Ferguson-Forcade Algorithm, Integer Relation, Schanuel's Conjecture

References

Bailey, D. H. ``Numerical Results on the Transcendence of Constants Involving $\pi$, $e$, and Euler's Constant.'' Math. Comput. 50, 275-281, 1988.

Sackell, J. ``Zero-Equivalence in Function Fields Defined by Algebraic Differential Equations.'' Trans. Amer. Math. Soc. 336, 151-171, 1993.